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User blog:Eners49/New notation idea?
Here is the basics: *{a} = a!!!...!!! with a nested factorial signs. Pretty basic for 1-entry arrays. That's already a really powerful function. Just a single factorial sign is powerful by the layman's standards, but if we have a bunch of them, that's going to be amazing! *{0} = 0 *{1} = 1! = 1 *{2} = 2!! = 2! = 2 *{3} = 3!!! = 6!! = 720! = 10^1,746.42 *{4} = 4!!!! = 24!!! = 10^10^10^24 *In general, {a} is about 10^^a With one-entry arrrays, we already have created a strong function, but we're only in between f2(n) and f3(n) in the fast-growing hierarchy, so we need to keep going. Let's see what two-entry arrays look like: *{a, b} = }...}}}, b-1} where there are {a, b-1} brackets. We've created a pretty strong recursion so far. *{a, 1} = {a}. Just like BEAF, if there are any 1's at the end, they can be removed. *{1, 2} = }, 1} = }} = } = *{4, 2} = }}, 1} = }}} = }} = 10^^10^^10^^10^^(10^10^10^24) *In general, {a, 2} is about 10^^^a Not too bad - we've already achieved a pentation level function. But that's still only between f3(n) and f4(n) in the fast-growing hierarchy. *{3, 3} = }, 2} = {10^^10^^10^^(10^1,750), 2} = }, 1} = about 10^^^10^^^(10^1,750) *In general, {a, 3} is about 10^^^^a *In general, {a, b} = 10^^^...^^^a with b+1 arrows Wow! With just two entries in our little arrays, we have created something that grows as fast as Knuth's arrows do. Thus, we have hit growth rate ω in the fast-growing hierarchy, starting from scratch! This is as powerful as the Ackermann function and three-entry BEAF. But the three-entry arrays grow faster quicker. *{a, b, c} = }...}}}, b-1, c} with a bracket sets *{a, 1, c} = {a, a, c-1} So {a, b, 2} has growth rate ω2, {a, b, 3} has growth rate ω3, etc. so the limit of 3-entry arrays is ω2. That's the same growth rate as 4-entry arrays in BEAF! So far, we're one entry ahead of BEAF. With 4+ entries, it's pretty much the same as normal BEAF (thanks for the suggestion Syst3ms): # signifies the rest of the array. *{a, 1, ..., 1, c+1, #} = {a, 1, ..., a, c, #} So 4-entry arrays have growth rate ω3, 5-entry arrays have growth rate ω4, etc., and a-entry arrays have growth rate ωa-1. So we can say that the limit of this is ωω. But beyond this, it's a little different. *{a | b} = {a, a, a, ..., a, a} with b a's *{a, 2 | b} = the following: *Stage 1 = {a | b} = {a, a, a, ..., a, a} *Stage 2 = {{a | b} | b} = {a, a, a, ..., a, a} with Stage 1 a's *Stage 3 = {{{a | b} | b} | b} = {a, a, a, ..., a, a} with Stage 2 a's *... *{a, 2 | b} = Stage a Insane, right?? {a, 2 | b} has growth rate ω2ω. *{a, 3 | b} = the following: *Stage 1 = {a, 2 | b} *Stage 2 = {{a, 2 | b}, 2 | b} *Stage 3 = {{{a, 2 | b}, 2 | b}, 2 | b} *... *{a, 3 | b} = Stage a {a, 4 | b}, {a, 5 | b}, etc. should be obvious. Currently, the limit of this is ω^ω^2. But we need to add 3 entries on the left now! *{a, b, 2 | c} = the following: *Stage 1 = {a, b | c} *Stage 2 = {a, {a, b | c} | c} *Stage 3 = {a, {a, {a, b | c} | c} | c} *... *{a, b, 2 | c} = Stage b *{a, b, 3 | c} = the following: Crazy, right? *Stage 1 = {a, b, 2 | c} *Stage 2 = {a, {a, b, 2 | c}, 2 | c} *Stage 3 = {a, {a, {a, b, 2 | c}, 2 | c}, 2 | c} *... *{a, b, 3 | c} = Stage b That's insane, right?? The limit of this is ω^ω^3. That's not too bad! We can generalize the pattern we have been making so far to have four, five, etc. entries on the left. In general, a entries on the left has a growth rate of ω^ω^a, so we can say that the maximum growth rate of this is ω^ω^ω, or ω^^3. Take that, ω^ω^3! Now, we will define: *{a | b, 2} = {a, a, a, ..., a, a | b} with b a's. *{a, 2 | b, 2} = the following: *Stage 1 = {a | b, 2} *Stage 2 = {{a | b, 2} | b, 2} *Stage 3 = {{{a | b, 2} | b, 2} | b, 2} *... *{a, 2 | b, 2} = Stage a (WHAT DO I PUT HERE? a or b) {a, 3 | b, 2}, etc. should be obvious. That's it so far, but I promise, I will add more soon! Please, someone who is skilled with ordinals and FGH, TELL ME THE MAXIMUM GROWTH RATE of {a, b | c, 2}. Category:Blog posts